Foundation Rs Wave Function Collapse

Collapse as ledger posting

Wave function collapse is the abrupt change a quantum system undergoes when it is measured: a spread-out set of possibilities becomes one definite outcome. In Recognition Science, the framework models this event as a ledger, a discrete record of events, posting an entry. The measurement is a recognition event, and the framework's eight-tick cycle sets the minimal time such an event can take. Collapse is then the system minimizing its cost, the forced price of recognition, to the nearest rung on a phi-power ladder. No hidden variables are needed in this account; the ledger itself supplies the definiteness.

The module under this page, RS_Wave_Function_Collapse.lean, is more modest than its title. It defines a cost function domainCost(m, e) = Jcost(m / e), where Jcost is the forced cost function from the framework's core theorem. The machine-checked library of formal theorems proves three general facts about this cost. First, when the measured value m equals the expected value e, the cost is zero. Second, for positive m and e, the cost is never negative. Third, the threshold phi minus 3/2, where phi is the golden ratio, is positive. These facts hold for any positive real numbers; the module proves nothing specific to wave function collapse itself.

The reason is that the module never defines what m and e mean for a quantum system. The docstring records the intended research direction, but the formal content is a template shared verbatim with 2383 sibling modules. The universal version lives in DomainCostTemplate.lean. To turn this into a theorem about collapse, one would need a definition of m and e in the subject's own terms, for instance a wave function's amplitude and its measured value. Without that, the module is a skeleton: the cost facts are proved, but the physical bridge is open.

What the module does establish, in plain language, is that the framework's cost function has the basic properties a ledger entry needs. A posting that matches expectation costs nothing, a posting never has negative cost, and the golden-ratio threshold is a real, positive quantity. These are the groundwork for a collapse story, not the story itself. The reader can now see exactly where the framework stands: the cost machinery is in place, and the missing piece is a definition that ties it to quantum measurement.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty RSWFCollapseCert := ⟨cert⟩

What this page does not claim

This module proves that wave function collapse occurs or how it works physically. The framework's collapse model is not a derivation from the cost function alone. No hidden variables are needed is a research note, not a proved result in this module.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/RS_Wave_Function_Collapse.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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